First order logic is the strongest logic satisfying both the Lowenheim-Skolem theorem and the theorem named for this property, according to Lindstrom's theorem. A simple case 
First order logic is the strongest logic satisfying both the Lowenheim-Skolem theorem and the theorem named for this property, according to Lindstrom's theorem. A simple case of that theorem for propositional logic can be proven by showing that Stone spaces have this property. That version of the theorem says that a theory is satisfiable if and only if every finite subset of that theory is satisfiable, and can be proven using Tychonoff's theorem. A real valued function on this type of space attains a maximum, according to the extreme value theorem. The sequential version of this property holds for a set in which all sequences have a convergent subsequence, which in R n is equivalent, by the Bolzano-Weierstrass theorem, to being closed and bounded. For ten points, the Heine-Borel theorem gives a characterization of this property in R n which says that any open cover has a finite subcover.
First order logic is the strongest logic satisfying both the Lowenheim-Skolem theorem and the theorem named for this property, according to Lindstrom's theorem.
{compactness}